Insights into the quantification and reporting of model-related uncertainty across different disciplines.

Quantifying uncertainty associated with our models is the only way we can express how much we know about any phenomenon. Incomplete consideration of model-based uncertainties can lead to overstated conclusions with real-world impacts in diverse spheres, including conservation, epidemiology, climate science, and policy. Despite these potentially damaging consequences, we still know little about how different fields quantify and report uncertainty. We introduce the "sources of uncertainty" framework, using it to conduct a systematic audit of model-related uncertainty quantification from seven scientific fields, spanning the biological, physical, and political sciences. Our interdisciplinary audit shows no field fully considers all possible sources of uncertainty, but each has its own best practices alongside shared outstanding challenges. We make ten easy-to-implement recommendations to improve the consistency, completeness, and clarity of reporting on model-related uncertainty. These recommendations serve as a guide to best practices across scientific fields and expand our toolbox for high-quality research.

Restricted Boltzmann Machines as Models of Interacting Variables.

We study the type of distributions that restricted Boltzmann machines (RBMs) with different activation functions can express by investigating the effect of the activation function of the hidden nodes on the marginal distribution they impose on observed binary nodes. We report an exact expression for these marginals in the form of a model of interacting binary variables with the explicit form of the interactions depending on the hidden node activation function. We study the properties of these interactions in detail and evaluate how the accuracy with which the RBM approximates distributions over binary variables depends on the hidden node activation function and the number of hidden nodes. When the inferred RBM parameters are weak, an intuitive pattern is found for the expression of the interaction terms, which reduces substantially the differences across activation functions. We show that the weak parameter approximation is a good approximation for different RBMs trained on the MNIST data set. Interestingly, in these cases, the mapping reveals that the inferred models are essentially low order interaction models.

On the Complexity of Logistic Regression Models.

We investigate the complexity of logistic regression models, which is defined by counting the number of indistinguishable distributions that the model can represent (Balasubramanian, 1997). We find that the complexity of logistic models with binary inputs depends not only on the number of parameters but also on the distribution of inputs in a nontrivial way that standard treatments of complexity do not address. In particular, we observe that correlations among inputs induce effective dependencies among parameters, thus constraining the model and, consequently, reducing its complexity. We derive simple relations for the upper and lower bounds of the complexity. Furthermore, we show analytically that defining the model parameters on a finite support rather than the entire axis decreases the complexity in a manner that critically depends on the size of the domain. Based on our findings, we propose a novel model selection criterion that takes into account the entropy of the input distribution. We test our proposal on the problem of selecting the input variables of a logistic regression model in a Bayesian model selection framework. In our numerical tests, we find that while the reconstruction errors of standard model selection approaches (AIC, BIC, ℓ1 regularization) strongly depend on the sparsity of the ground truth, the reconstruction error of our method is always close to the minimum in all conditions of sparsity, data size, and strength of input correlations. Finally, we observe that when considering categorical instead of binary inputs, in a simple and mathematically tractable case, the contribution of the alphabet size to the complexity is very small compared to that of parameter space dimension. We further explore the issue by analyzing the data set of the "13 keys to the White House," a method for forecasting the outcomes of US presidential elections.